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Pfaffian and fragmented states at ν =52

in quantum Hall droplets

H. Saarikoski,1 E. Tolo,2 A. Harju,2 and E. Rasanen3, 4

1Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, the Netherlands∗2Helsinki Institute of Physics and Department of Applied Physics,

Helsinki University of Technology, P.O. Box 4100, FIN-02015 HUT, Finland3Institut fur Theoretische Physik, Freie Universitat Berlin, Arnimallee 14, D-14195 Berlin, Germany

4European Theoretical Spectroscopy Facility (ETSF)

When a gas of electrons is confined to two dimensions, application of a strong magnetic fieldmay lead to startling phenomena such as emergence of electron pairing. According to a theory thismanifests itself as appearance of the fractional quantum Hall effect with a quantized conductivity atan unusual half-integer ν = 5

2Landau level filling. Here we show that similar electron pairing may

occur in quantum dots where the gas of electrons is trapped by external electric potentials into smallquantum Hall droplets. However, we also find theoretical and experimental evidence that, dependingon the shape of the external potential, the paired electron state can break down, which leads to afragmentation of charge and spin densities into incompressible domains. The fragmentation of thequantum Hall states could be an issue in the proposed experiments that aim to probe for non-abelianquasi-particle characteristics of the ν = 5

2quantum Hall state.

PACS numbers: 73.21.La, 73.43.-f, 71.10.Pm, 85.35.Be

I. INTRODUCTION

The discovery of the quantum Hall (QH) effect at Lan-dau level filling factor ν = 5

2in the two-dimensional elec-

tron gas (2DEG) (Ref. 1) marked evidence that incom-pressible states may form at unusual even-denominatorfilling fractions. After years of subsequent theoreticaland experimental work2,3,4,5 it is well established thatone of the most plausible theoretical candidates for aQH state at ν = 5

2is an exotic state of matter, a paired

quantum Hall state. Since electron-electron (e-e) inter-actions are repulsive this pair formation is a collectivephenomenon involving residual interactions of compositeparticles that, in this state, are composites of an electronand two vortices. The electron pairing would be analo-gous to the formation of Cooper pairs in superconduc-tors, although it would be purely a result of e-e interac-tions without contribution from phonons or other fields.In some theoretical models, the excitations of the pairedelectron state are predicted to have non-abelian statisticsthat could be employed in the field of topological quan-tum computing.2 Currently, the most pressing challengeis to experimentally find evidence of the paired electronstate and the particle statistics of its excitations.6,7,8 Theproposed tests9 for the non-abelian properties of theseexcitations make use of confined geometries and multi-ple constrictions in the 2DEG to generate interferenceamong tunneling paths. This leads to a natural ques-tion whether the paired electron state is stable when the2DEG is confined into narrow trappings.

This work addresses the structure of the ν = 52

statewhen electrons in the 2DEG have been confined by exter-nal potentials into small QH droplets. They can be ex-perimentally realized by placing semiconductor quantum-dot (QD) devices into strong magnetic fields.10 We showhere theoretical evidence that in QH droplets the Pfaf-fian wave function,3 which is commonly used to describe

electron pairing, may have high overlaps with the exactmany-body states at ν = 5

2. In these calculations, we

assume that the half-filled Landau level is spin-polarizedand use realistic e-e potentials that include screening ef-fects from the background charge of electrons in the thelowest Landau level and a softening due to the finitethickness of the sample. However, the half-filled secondLandau level of the Pfaffian state has a relatively highangular momentum, which may lead to its instability inthe QD confinement. We show that in harmonic confin-ing potentials a compact filling of the half-filled Landaulevel is favored leading to the lowering of its angular mo-mentum. The paired electron state would then breakdown via fragmentation of spin and charge densities intotwo incompressible domains, spin-compensated ν = 2 atthe edges and spin-polarized ν = 3 at the center (see Refs.11 and 12). This phenomenon is analogous to the pro-posed formation of similar structures in the 2DEG wheretranslational symmetry has been broken by long-rangedisorder.13 We present the fragmented states in QDs asalternatives to the Pfaffian state and show signatures ofthem in electron transport experiments. Based on theseresults, we conjecture that the stability of the paired elec-tron state depends crucially on the shape of the potentiallandscape where the electrons move in the 2DEG. Thismay explain, e.g., the observed fragility of the ν = 5

2QH

state in narrow quantum point contacts.14

The paper is organized as follows. We introduce ourtheoretical model of QDs in Sec. II and the computationalmethods used to solve the many-body problem in Sec. III.The exact diagonalization method is used in Sec. IV tocalculate the overlaps of the Pfaffian wave function withthe exact many-body state. In Sec. V, we analyze theelectronic structure of fragmented QH states and showthat the second-lowest Landau level is spin-polarized dueto the lifting of degeneracy of single-particle states nearthe Fermi level. In Sec. VI, we present experimental ev-

2

idence for fragmentation of QH states in the 2 ≤ ν ≤ 52

filling-factor regime. Section VII concludes our work withdiscussion of the relevance of our findings with the ob-served fragility of the ν = 5

2QH state in disordered or

confined 2DEG.

II. MODEL

QDs formed in the GaAs/AlxGa1−xAs heterostructureare modeled for both lateral and vertical QD devices asdroplets of electrons in a strictly two-dimensional (2D)plane confined by a parabolic external potential.10 Weuse an effective-mass Hamiltonian

H =N∑

i=1

[

(pi + eA)2

2m∗+ Vc(ri)

]

+e2

4πǫ

∑

i<j

1

rij

, (1)

where N is the number of electrons, Vc(r) = m∗ω20r

2/2 isthe external parabolic confinement, m∗ = 0.067me is theeffective mass and ǫ = 12.7ǫ0 is the dielectric constantof the GaAs semiconductor medium, and A is the vectorpotential of the homogeneous magnetic field B perpen-dicular to the QD plane. The confinement strength ω0

in the calculations is 2 meV, unless otherwise stated.If the e-e interactions are excluded, the single-

particle solutions of the Hamiltonian (1) are Fock-Darwinstates.15 In the limit of a very high magnetic field, theLandau level structure approaches that of the 2DEG.However, in finite magnetic fields the external poten-tial alters the electronic structure and different Lan-dau levels overlap. Therefore, the concept of Landaulevel filling needs to be generalized to finite-size sys-tems. Kinaret et al. defined the average filling factoras νavg = N2(N + L)/2, where L is the total angularmomentum.16 Another possibility is to focus on the low-est Landau level (LLL) and define filling factor of a stateas νLLL = 2N/NLLL. These definitions differ in the highfilling-factor regime, but this is not critical to the in-terpretation of results that are based on the structuralproperties of the many-body states.

III. COMPUTATIONAL MANY-BODY

METHODS

The ground state corresponding to interacting elec-trons in QH droplets is solved numerically using the exactdiagonalization (ED), density-functional theory (DFT),and the variational quantum Monte Carlo method(QMC). Since the paired electron state in the 2DEG is astrongly correlated many-body state, the ED method isused to analyze its stability in the QD confining poten-tial. The DFT and QMC methods are used to analyzethe fragmented QH states. The regime where this frag-mentation gives characteristic signals in the experimentsis beyond the reach of the ED method. However, we find

that both the DFT and QMC methods provide accurateresults in this regime (see Appendix).

A. Exact diagonalization

In the ED method, we assume that the electrons oc-cupy states on one Landau level only. If we now take afixed number of states from this Landau level, our com-putational task is first to construct the many-body basis.Then the Hamiltonian matrix corresponding to Hamilto-nian of Eq. (1) is constructed in this basis. Finally, thelowest eigenstate and eigenvalue are found by matrix di-agonalization. More details can be found, e.g., in Ref. 17.In addition to the standard Coulomb interaction, we usein the ED two modifications of it. To model the finitethickness of the sample, we use a softened potential18

defined as

VT (r) =e2

4πǫ√

r2 + d2T

, (2)

where dT is the sample thickness. Electrons in second orhigher Landau levels move on top of background chargeof lower Landau levels, which effectively screens theCoulomb interaction. This is modeled with a screenedpotential that is of the Gaussian form

VS(r) =e2 exp(−r2/d2

S)

4πǫr, (3)

where dS is the screening length. The unit of lengthin our ED results is given by l =

√

~/m∗ω, where

ω =√

ω20 + (ωc/2)2 and ωc = eB/m∗ is the cyclotron

frequency of electron in magnetic field B.

B. Density-functional theory

Our DFT approach is based on spin-DFT, a variant ofthe conventional DFT generalized to deal with non-zerospin polarization. On top of standard spin-DFT, we in-clude the bare external vector potential A [see Eq. (1)]in the Kohn-Sham equation. In contrast with current-spin-DFT, however, we neglect the exchange-correlationvector potential Axc. In the magnetic-field range con-sidered here, this has been shown to be a very reason-able approximation.19 As another valid approximation,we neglect the dependence of the exchange and corre-lation on the vorticity.20 The exchange and correlationenergies and potentials are calculated using the 2D localspin-density approximation, for which we use the QMCparametrization of the correlation energy by Attaccaliteet al.21

The DFT approach is implemented on a 2D real-space grid and employs a multigrid method for solving ofthe Kohn-Sham equations.22 Our symmetry-unrestrictedDFT approach has been shown to lead to solutions with

3

broken rotational symmetry that has been linked to mix-ing of the different eigenstates of angular momentum.23,24

In a fixed symmetric external potential, this type of spon-taneous symmetry breaking is expected to be unphysical.In Sec. VI, we compare the validity of this assumptiondirectly to experimental data.

C. Quantum Monte Carlo

Since the fragmentation of many-body state in thevicinity of ν = 5

2is a delicate many-body problem, we

employ the QMC method to analyze the reliability of ourDFT approach. The wave function in the QMC is chosento be

Ψ = D↑D↓

N∏

i<j

J(rij) , (4)

where the two first factors are Slater determinants for thetwo spin types, and J is a Jastrow two-body correlationfactor. The Slater determinants are constructed from theFock-Darwin states. For the two-body Jastrow factor, weuse a form

J(r) = exp

(

Cr

a + br

)

, (5)

where a is fixed by the cusp condition to be three fora pair of equal spins and one for opposite ones, andb is an additional parameter different for both spin-pair possibilities. The ground state of the QD in thespin-droplet regime is calculated assuming that the LLLand the second-lowest Landau level (SLL) are com-pact. This means that the Slater determinants arebuilt from single-particle states having angular momental = 0, . . . , NLLL,s − 1 for spin s =↑, ↓ in the LLL andl = −1, . . . , NSLL,s − 2 for the spin s in the SLL. En-ergy for each combination of non-negative total spin Sand total angular momentum L is then calculated. TheQMC method deals with the correlation effects in themany-particle system more accurately than the DFT ap-proach. However, the computational cost of the QMCis significantly larger than that of the DFT. A detaileddescription of the QMC method is given in Ref. 25.

IV. PFAFFIAN STATE IN QUANTUM DOTS

The structure of the QH states in the 2DEG at half-integer filling factor has been a topic of intense researchefforts.2 Currently, it is regarded plausible that the ex-perimentally observed ν = 5

2state consists essentially

of a full spin-compensated LLL and a half-filled spin-polarized SLL,5 in which weak p-wave electron pairingtakes place. Formally, the SLL is described by a Moore-Read, or Pfaffian, wave function lifted to the SLL.3,4

There exists some theoretical evidence that the excita-tions of this QH state obey non-abelian statistics.2,3,26

ED calculations have become standard tests of trial wavefunctions of QH states, and they have shown high over-laps with the Pfaffian wave function in the 2DEG.27

However, there are other candidates for the ν = 52

state, some of which possess only abelian quasiparticleexcitations.26,28

The structure of the ν = 12

state in QDs was analyzedwith the ED method in Ref. 11. Here we provide resultsfor half-filled higher Landau levels with more realisticinter-electron potentials defined in Sec. III. Following thetheory of the ν = 5

2QH state in the 2DEG, we assume

that the half-filled Landau level is spin-polarized. ThePfaffian wave function,3 which describes paired fermionstates of the half-filled Landau level, is defined for LLLas

ΨPF = Pf

(

1

zi − zj

)

∏

i<j

(zi−zj)2 exp

(

−1

2

∑

i

r2i

)

. (6)

In higher Landau levels the Pfaffian state is obtainedby applying the Landau level raising operator to eachelectron. The angular momentum of the Pfaffian state isL′ = N ′(N ′ − 1)− (nLL + 1

2)N ′, where N ′ is the number

of electrons in the half-filled Landau level and nLL ={0, 1, . . .} is the Landau level index.

We present the overlaps of the Pfaffian wave functionwith the ED eigenstate for electrons frozen to lowest(LLL), second (SLL), or third (TLL) Landau level, whichcorrespond to filling fractions of ν = 1

2, ν = 5

2, and ν = 9

2,

respectively. Electrons in the half-filled second and thirdLandau levels move on top of the uniform backgroundelectron density of the spin-compensated lower Landaulevels. This background charge effectively screens theCoulomb interaction. In QDs, the e-e interactions arefurther screened due to metallic leads.

Figure 1 shows the overlaps of the Pfaffian wave func-tion and the ED eigenstate of Coulomb interaction forparticle numbers 4 ≤ N ′ ≤ 12. For large particle num-bers, the overlaps in the second Landau level are highest.This shows that ν = 5

2has the highest probability to be

described by the Pfaffian.Next, we study how the screening of the e-e interaction

and finite thickness of the sample change the overlaps ofthe ED eigenstate with the Pfaffian. For six electrons onLLL, the overlaps are slightly improved when the screen-ing and finite sample thickness are taken into account inthe interaction (see Fig. 2). On SLL, screening slightlyimproves the overlap, but a finite thickness lowers it. Thesame trends can be seen in Fig. 2 for eight electrons, butnow the effects are clearly stronger, and there is a largeincrease in the overlaps. On the LLL, a finite samplethickness is needed to achieve the best overlap. On theSLL, the screening increases the overlap, which can becontrasted with the spherical geometry where the SLLoverlap is maximized at a finite thickness of the sample.29

The highest overlaps are of the order of 0.8–0.9 atν = 5

2, which means that the structure of the many-

body state is close to the Pfaffian. The exact state at the

4

4 5 6 7 8 9 10 11 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

N’

Ove

rlap

LLLSLLTLL

FIG. 1: (Color online) The overlaps of the Pfaffian wave func-tion with the corresponding exact state at the lowest (LLL),second (SLL), and third (TLL) Landau level in the case ofa Coulombic electron-electron interaction. N ′ denotes thenumber of electrons in the half-filled Landau level.

given angular momentum would therefore show electronpairing to a significant degree. We note that the Pfaffianwave function in Eq. (6) has no adjustable parameters. Itis possible to modify the Pfaffian wave function by intro-ducing a pairing function that differs from g = 1/(zi−zj)of the Moore-Read form.30 This won’t change the angularmomentum of the state but has been found to increaseoverlaps significantly in the 2DEG.

In addition to the overlaps, it is crucial to verify thatthe state at the angular momentum of the Pfaffian stateis energetically favorable. In fact, the LLL ν = 1

2state

with N ′ = 8 corresponding to Fig. 2(d) is a possibleground state at small and large values of the thicknessdT , but not at values of dT where the overlap is peaked.31

A further obstacle for the Pfaffian state in finite-size QHdroplets is that the SLL may not attain the high angularmomentum and complete spin polarization of the Pfaf-fian. In QH droplets, the degeneracy of Landau levels islifted when electrons move in external confining poten-tials [Fig. 3(b)], and a compact distribution of electronson the Landau levels could be energetically more favor-able. In the next section, we show that this would leadto nonexistence of the paired electron state and intro-duce fragmented QH states in quantum Hall droplets asalternatives.

V. FRAGMENTED QUANTUM HALL STATES

In quantum Hall droplets, single-particle states withineach Landau level are not degenerate due to the confin-ing potential. The average distance of an electron fromthe center of the droplet, and therefore also the potentialenergy, increase with angular momentum. This suggests

that a compact occupation structure may be energeti-cally favorable. The compact occupation of Landau lev-els leads to fragmentation of charge and spin densitiesinto incompressible integer filling factor domains. Wecall these states fragmented quantum Hall states that arealternatives to the paired electron state at half-integerLandau level fillings.

We analyze the structure of fragmented QH states nearν = 5

2in a harmonic confining potential of a semicon-

ductor quantum dot with the QMC and the DFT meth-ods. The Kohn-Sham single-particle energy spectrum ofthe Landau levels calculated with the spin-compensatedDFT and the spin-DFT are shown in Fig. 3(a) and (b),respectively. The spin-DFT and the QMC show that thedegeneracy of the single-particle states close the Fermienergy is lifted via a complete polarization of the second-lowest Landau level. Therefore, a compact occupationof the single-particle states of the spin-compensated LLLand spin-polarized SLL leads to a fragmented state witha ν = 2 region (double-occupied LLL) at the edges ofthe droplet and ν = 3 (spin-polarized SLL) at the center[Fig. 4].

The spin-splitting of the SLL in the spin-DFT calcu-lations is analogous to the Stoner criterion, which statesthat in the presence of correlations between electrons ofthe same spin and high density of states near the Fermilevel, the system prefers ferromagnetic alignment thatreduces the degeneracy.32 We call the incompressible,spin-polarized droplet of SLL electrons a spin droplet.12

The size of the spin-droplet becomes significant when thenumber of electrons in the dot N & 35. The non-uniformfilling-factor structure of the spin-droplet states is rem-iniscent of the incompressible QH domains that form inthe 2DEG with long-range disorder.13 The compact oc-cupation of the SLL leads to a lower angular momentumthan what is needed for a paired electron state as de-scribed by the Pfaffian wave function (6). For example,the size of the spin-droplet in the QMC method is N ′ = 8electrons at N = 48, and the angular momentum of theSLL is L′ = 20, which can be contrasted to L′ = 44for the Pfaffian wave function with the same number ofelectrons.

The SLL remains polarized and compact between 52≥

ν ≥ 2. Hence, we call this filling-factor range the spin-droplet regime. The size of the spin-droplet graduallyshrinks with the increasing magnetic field as the electronsare passed from the SLL to the LLL. The contributionsof the LLL and SLL occupancies to the electron and spindensities are shown in Fig. 4 for the case of 60-electronQD. Qualitatively similar results were obtained for con-finement strengths 1 to 4 meV and electron numbers Nbetween 35 and 120, which confirms the generality of theresults. The calculations show that the energy benefitfrom the polarization of the SLL is large (see Fig. 3 andAppendix), which would make spin-droplets robust in thepresence of impurities in samples.

We note that the stability of the fragmented QH statesin large QDs (N > 30) can be contrasted to the insta-

5

0 1 2 3

0.7

0.8

0.9

Ove

rlap

1/dS (1/l)

a)

N’=6

0 1 2 30.4

0.5

0.6

0.7

0.8

0.9

Ove

rlap

1/dS (1/l)

N’=8

b)

LLLSLL

0 2 4 6 8 10

0.7

0.8

0.9

Ove

rlap

dT (l)

c)

N’=6

0 2 4 6 8 100.4

0.5

0.6

0.7

0.8

0.9

Ove

rlap

dT (l)

d)

N’=8

FIG. 2: Overlap of the Pfaffian wave function with the corresponding exact state for N ′ = 6 and N ′ = 8 electrons in the half-filled Landau level, respectively, using (a-b) screened electron-electron potential with screening length dS and (c-d) softenedpotential due to finite sample thickness dT for electrons at the lowest Landau level (LLL) corresponding to ν = 1

2, and the

second lowest Landau level (SLL) corresponding to ν = 5

2.

bility of the maximum-density-droplet (MDD) state inthe same regime. The MDD state is the totally polar-ized state corresponding to the ν = 1 QH state in 2DEG,and it has been found to be instable in large QDs withN > 30 (Refs. 33 and 34).

VI. SIGNATURES OF FRAGMENTATION IN

ELECTRON AND SPIN TRANSPORT

The emergence of finite size counterparts of integer andfractional QH states in QDs gives characteristic signa-tures in the chemical potentials. Several experimentalmethods have been developed to measure the chemicalpotential in a QD via addition of electrons one-by-oneinto the system. These experimental methods includeCoulomb blockade,35 capacitance,36 and charge detec-tion techniques.37 In this work, we use data from elec-tron transport measurements of QDs in the Coulomb andspin blockade regime.34,38 The spacings of the spin andCoulomb blockade peaks correspond to the energy neededto add the Nth electron in the system of N − 1 elec-trons, i.e., the chemical potential defined as µ(N, B) =Etot(N, B) − Etot(N − 1, B).

We calculate the signatures in the chemical potentialsassociated with the formation of fragmented QH states

and compare these to those obtained from the electrontransport data in three different QD devices. Two ofthe experimental samples (sample A and B) are lateralquantum dots on a high-mobility 2DEG38 while the thirdone (sample C) is a vertical QD.34 The samples A andB were manufactured on a high mobility 2DEG sampleswith spin-polarized leads for electron transport measure-ments in the spin blockade regime. The data of the sam-ple C was obtained in the Coulomb blockade. The highmobility of samples chosen for comparison is essential toreduce unpredictable effects of impurities and disorderthat make identification of signals of physical phenom-ena difficult.

We first address the problem of whether the electronicstates in the QD samples show any signs of broken rota-tional symmetry. Inhomogeneities and impurities in QDdevices may break the rotation symmetry, and a Jahn-Teller type of mechanism could be active if disorder alterssignificantly the shape of the confining potential. As aresult, the ground state transitions with increasing mag-netic field become continuous rather than discrete. Asignature of this type of symmetry breaking would be asmoothing of the chemical potential. Experimental datafrom a high-mobility lateral QD device is of sufficientlygood quality to test for the presence of symmetry break-ing mechanisms. Figure 5 shows a comparison of the elec-

6

0 10 20

120

122

124

|L|

Ene

rgy

(meV

)

0 10 20

120

122

124

|L|

Ene

rgy

(meV

)

spin .spin .

b) FEF E

LLL

second Landau level(SLL)

a)

spin polarizedSLL

degeneracy spin splitting

lowest Landau level (LLL)

FIG. 3: (Color online) (a) Kohn-Sham energy spectrum of a60-electron quantum dot as a function of single-particle angu-lar momentum L calculated from the density functional the-ory with spin-compensated orbitals. The density of states ofthe second-lowest Landau level (SLL) is high near the Fermienergy EF. The magnetic field is 2.125 T which correspondsto a filling factor of ν = 5

2. (b) The corresponding energy

spectrum from spin-density functional theory shows lifting ofthe degeneracy near the Fermi level via complete spin polar-ization of the SLL. The lowest Landau level (LLL) remainsspin compensated. Spin ↑ (↓) corresponds to spin orientationparallel (antiparallel) to the magnetic field. The spin-splittingdue to many-body effects is about 1.5 meV at L = 0. In com-parison, the Zeeman splitting is about 0.05 meV.

300 nm(a)

(c)

ρ

ν = 5/2

ν = 2

(b)

2 < ν < 5/2

FIG. 4: (Color online) Total electron density ρ↑ + ρ↓ (fullregion) and the net spin density ρ↑ − ρ↓ (transparent blueregion) of quantum Hall states in a quantum dot at (a) ν = 2,at (b) an intermediate state between ν = 2 and ν = 5

2, and at

(c) ν = 5

2. The latter two show fragmented charge and spin

densities with spin-compensated ν = 2 region at the edgesand spin-polarized ν = 3 at the center. The densities werecalculated with the spin-density-functional theory for a 60-electron quantum dot. The net spin-up density is due to spinpolarization of the second-lowest Landau level.

tron transport data to DFT calculations with and with-out symmetry breaking. The data shows sharp increasesin the chemical potentials, which is consistent with dis-crete transitions in the ground state. Therefore, to a goodapproximation, the rotational symmetry is preserved inhigh mobility samples, and the angular momentum L isa good quantum number.

The complete polarization of the SLL at ν = 52

is re-flected in the energetics of the system. The DFT cal-culations show that at ν = 5

2there is a step feature

followed by a plateau region in the chemical potential.

1.2 1.4 1.6

0.5

1

1.5

E(N

=48)

−E(N

=47)

(ar

b. u

nits

)

B (T)

E(N

=47)

− 2

296

(meV

) L,S= 355, 5/2

375, 7/2

394, 7/2

409, 7/2

425, 5/2

443, 5/2

462, 3/2

483, 3/2

505, 1/2

506, 3/2

529, 1/2

ν=5/2 ν=2

rotation symmetry broken symmetry sample B

FIG. 5: (Color online) Ground state energy of the N = 47quantum dot calculated with the density-functional theory forthe rotationally symmetric eigenstates of the angular momen-tum L (blue curve) and for the ground states in the symmetry-unrestricted approach (magenta curve). The correspondingchemical potentials µ(N = 47 → 48) calculated from the the-ory are shown in the lower panel together with experimentaldata from a lateral quantum-dot device (sample B). Dashedlines correspond to the boundaries of the spin-droplet regime.The insets show the fragmented spin and charge densities ofthree of the corresponding states (cf. Fig. 4). The strengthof the parabolic confining potential of the quantum dot isω0 = 2 meV in the calculations.

Figure 6 shows the DFT results for chemical potentialsof N = 24 . . . 48 in comparison with the experiments.The step in the chemical potential is associated with thetotal polarization of the SLL in the DFT calculations.This feature can be found in all three samples we studiedabove N ≥ 30, as predicted by the theory. Some modelsof QDs assume that the SLL is spin-polarized due to theZeeman effect.39 This model does not, however, apply forthe lateral and vertical QD devices examined in this workwhere the effect of the Zeeman splitting is estimated tobe only a few percent of the spin splitting caused by themany-body interactions (see Fig. 3).

In the 52

> ν > 2 regime, the ground state energy isapproximately constant (see Appendix), and the calcu-lations show a phase transition in the system where twophases (ν = 2 and ν = 3) coexist, and the size of theν = 2 domain increases with the magnetic field. Thechemical potential does not continue to rise, but instead,it is oscillating around a constant value until ν = 2.This signature can be found in all the experimental sam-ples (see Fig. 6). All electron transport data presentedare thus consistent with the theoretical picture that the

7

1 1.5 2 2.5

N=39

N=38

N=30

N=24

B (T)0.5 1 1.5

N=24

N=30

N=44

N=48

B (T)1.5 2 2.5 3

N=24

N=30

N=38

N=48

B (T)

(a) (b) (c)

µ(a

rb.u

nits

)

theory vertical device lateral device (sample B)

spin−dropletregime

ν=2ν=2

ν=2

ν=3

ν=3ν=3

ν=5/2 ν=2 ν=2ν=2ν=5/2ν=5/2ν=3

ν=3

ν=3

FIG. 6: (Color online) Chemical potentials calculated with the density-functional theory (a) and measured from a vertical(b) and lateral (c) quantum-dot devices for various electron numbers. Both experiments show the signal associated with thepolarization of the second lowest Landau level at ν = 5/2 in the peak position data when N & 30 in agreement with thetheoretical result. The confinement strength ranged from 2 to 4 meV depending on the electron number. The data for thevertical device in (b) is courtesy of L. Kouwenhoven,34 and the data for lateral device in (c) is courtesy of A. S. Sachrajda.38

ground states in the vicinity of ν = 52

involve fragmentedQH states. We point out, however, that the results aresensitive to the shape of the external potential, and thepairing of the electrons may still occur if the potential issufficiently homogeneous, e.g., in large QDs, where thesecond Landau level would acquire higher angular mo-mentum.

Spin polarization of the leads is commonly used to cre-ate a current that depends on the orientation of the elec-tron spin, which passes through the device. In the caseof the two lateral QDs in our analysis, the electrons en-ter the QD from spin-polarized magnetic edge states ofthe 2DEG through tunneling barriers. Coulomb block-ade lifts when the energies of the many-body states cor-responding to N and N + 1 electrons are equal. Thetunneling current depends then on the coupling betweenthe wave function in the QD and the electronic statesin the external leads. The lowest Landau level orbitalsare at the edges of the QD, and the coupling is strongerto the leads compared to the second lowest Landau levelorbitals that are close to the center of the QD. Due topolarization of the leads, their coupling to electron stateswith spin parallel to the external polarization is higherthan the coupling of spins antiparallel to the externalpolarization. This spin dependence in the transport hasbeen shown to lead to a characteristic checkerboard pat-tern of current densities through QDs.38,40,41,42,43 OurDFT results are consistent with such transport currentsin the spin blockade regime (Fig. 7). The polarization ofthe SLL in the 5

2≥ ν ≥ 2 regime would be in contrast to

the model presented in Ref. 40. A consequence of this is

that the transport current via SLL orbitals should showno checkerboard pattern in this regime since the spinsare always parallel to the polarization of the leads. Thiscould be tested with high-accuracy spin blockade spec-troscopy which would be able to detect small changes inthe weak tunneling currents through the SLL orbitals.

VII. FRAGILITY OF THE ν = 5

2QUANTUM

HALL STATE

The ν = 52

state is one of the most fragile QH states.It is observed only in high-mobility 2DEG samples as thepaired electron state may break down in the presence ofimpurities. These induce a non-uniform potential that, inthe light of results in this work, may lead to its instabil-ity. Our findings are thus in line with those obtained byChklovskii and Lee who predicted that in the presence oflong-range disorder in the 2DEG, incompressible integerfilling factor regions form that are separated by domainwalls.13 These structures are analogous to the fragmentedQH states that we find in QDs. Structures reminiscentof domain walls have been observed with scanning-probeimaging techniques in a perturbed QH liquid.44

Analogous instability of QH states may also occur inother geometries where the electrons are not strictly con-fined in all directions, such as in high-mobility 2DEGsamples in the vicinity of constrictions. One indicationof this may be the observed fragility of the ν = 5

2state

in narrow quantum point contacts.14 Proposed tests9 forthe non-abelian properties of quasi-particle excitations of

8

Experiments (Sample A)

N=47

N=45

N=46

curr

ent (

arb.

uni

ts)N=48

ν=2

B (T) 1.61.2ν=2

chem

ical

pot

enti

al(a

rb. u

nits

)

N=46

N=47

N=48

(high)

DFT calculation

(medium) (low)

LLL LLL SLL

FIG. 7: (Color online) Checkerboard pattern of transport current in density-functional theory (left panel) and spin-blockadeexperiments (right panel). The lowest current densities correspond to electron transport via states in the second-lowest Landaulevel, near the core of the quantum dot. The current density in experiments has been amplified in high magnetic fields with alinear function to compensate for the general attenuation of the signal.

ν = 52

QH state make use of finite geometries and mul-tiple constrictions to generate interference among tun-neling paths. A possible fragmentation of the ν = 5

2

QH state close to the boundaries, which would lead tothe instability in such geometries, is still an open ques-tion that requires further analysis of the effects of theconfinement. While recent experiments on the quasi-particle tunneling,6 shot noise generated by partitioningedge currents,7 and interferometric measurements of QHedge excitations8 of the ν = 5

2QH state show results,

which are all consistent with the unusual quasi-particlecharge e∗ = 1

4of the paired electron state, the parti-

cle statistics of the excitations remains to be confirmed.Possible fragmentation of QH states in narrow constric-tions needed for quasi-particle interferometry adds an-other challenge in this long quest to confirm the possiblenon-abelian characteristics of the ν = 5

2state.

To conclude, we have shown theoretical evidence thatelectron pairing is possible in small QH droplets in quan-tum dots at ν = 5

2, provided that the half-filled Lan-

dau level can acquire sufficiently high angular momen-tum. However, our calculations indicate that in parabolicexternal confining potentials the paired electron statebreaks down leading to fragmentation of charge and spindensities. We find indirect evidence of such fragmenta-tion in several experiments but point out that our resultscan be tested by direct measurements of the spatial de-pendence of spin and charge densities in different geome-tries and experimental setups.

Acknowledgments

We gratefully acknowledge valuable discussions withA. S. Sachrajda, M. Ciorga, S. M. Reimann, and L.Kouwenhoven, and thank Jaakko Nissinen for calcu-lating the Pfaffian interaction matrix elements. This

work was supported by the EU’s Sixth Framework Pro-gramme through the Nanoquanta Network of Excellence(NMP4-CT-2004-500198), the Academy of Finland, andthe Finnish Academy of Science and Letters through theViljo, Yrjo and Kalle Vaisala Foundation.

VIII. APPENDIX: ACCURACY OF

NUMERICAL METHODS

The electron correlations play an important role in thestructure of fractional QH states. To test for the accu-racy of the DFT method in the spin-droplet regime, wecompare the energies of different spin polarization statesbetween the DFT and the QMC in the 5

2≥ ν ≥ 2 regime.

The results for a 48-electron QD are shown in Fig. 8.Both methods show the spin-droplet structure with a

comparable energy benefit in the polarization δ ≈ 0.5meV for Smax = 4. The QMC method estimates that themaximum size of the spin droplet is NSD = 7 compared to8 in the DFT. Given the typical statistical error of ±0.05meV in the QMC results, the overall agreement betweenthe methods is excellent. This test indicates that theDFT method captures the essential many-body physics ofthe spin-droplet formation and gives accurate results forthe ground states. The DFT method was subsequentlyused in the calculation of the chemical potentials of largeQDs, which can be compared to the transport experi-ments in the spin blockade regime.

The DFT method predicts some non-compact statesoutside the spin-droplet regime, e.g., L = 375, S = 7/2state as shown in Fig. 5. This state has one spin-downelectron in the SLL with l = 0. Emergence of non-compact states is a manifestation of the degeneracy ofthe single-particle states near Fermi-level. However, theyare rare in the DFT and occur only at magnetic fieldsbelow the polarization of the SLL. Detailed analysis ofthese states with the QMC goes beyond the scope of the

9

E(N

=48)

- 2

260

(meV

)

S = 4S = 3S = 2

S = 0S = 1

N = 4814

12

10

8

61 1.2 1.4 1.6

B (T)

QMC

DFT

ν = 5/2

ν = 5/2

ν = 2

ν = 2

FIG. 8: (Color online) Comparison of the ground state en-ergies for given total spin S in the density-functional theory(DFT) and the quantum Monte Carlo (QMC) method. Thenumber of electrons N = 48. The line widths in the QMCdenote the statistical error in the results. Only the groundstate and the S = 0 state are shown in the DFT result. Thestrength of the parabolic confining potential of the quantumdot is ω0 = 2 meV in the calculations.

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